1/2^2 + 1/3^2 + 1/4^2 ... + 1/50^2 < 1
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NQ
0
T
0
K0
0
HL
1
11 tháng 4 2018
\(M=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{50^2}\)
\(M< \dfrac{1}{4}+\left(\dfrac{1}{2.3}+...+\dfrac{1}{49.50}\right)=\dfrac{1}{4}+M_1\)
\(M_1=\left(\dfrac{1}{2}-\dfrac{1}{3}\right)+\left(\dfrac{1}{3}-\dfrac{1}{4}\right)...+\left(\dfrac{1}{48}-\dfrac{1}{49}\right)+\left(\dfrac{1}{49}-\dfrac{1}{50}\right)\)
\(M_1=\dfrac{1}{2}+\left(-\dfrac{1}{3}+\dfrac{1}{3}\right)+...+\left(-\dfrac{1}{49}+\dfrac{1}{49}\right)-\dfrac{1}{50}=\dfrac{1}{2}-\dfrac{1}{50}\)
\(M< \dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{50}=\dfrac{3}{4}-\dfrac{1}{50}< \dfrac{3}{4}=>dpcm\)
LK
2
17 tháng 5 2016
Đặt A=1/2^2+1/3^2+1/4^2+...+1/50^2
A<1/1*2+1/2*3+1/3*4+...+1/49*50
A<1-1/2+1/2-1/3+1/3-1/4+...+1/49-1/50
A<1-1/50<1
Vậy A<1
BH
17 tháng 5 2016
Ta có:\(\frac{1}{2^2}< \frac{1}{1.2};\frac{1}{3^2}< \frac{1}{2.3};\frac{1}{4^2}< \frac{1}{3.4};...;\frac{1}{50^2}< \frac{1}{49.50}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)
mà \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}=1-\frac{1}{50}< 1\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}< 1\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}< 1\left(đpcm\right)\)
HT
1
20 tháng 4 2017
\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+...+\(\frac{1}{50^2}\)<1
ta có \(\frac{1}{2^2}\)<\(\frac{1}{1.2}\)
\(\frac{1}{3^2}\)<\(\frac{1}{2.3}\)
..........................
\(\frac{1}{50^2}\)<\(\frac{1}{49.50}\)
ta được \(\frac{1}{1.2}\)+\(\frac{1}{2.3}\)+...+\(\frac{1}{49.50}\)
=>1-\(\frac{1}{2}\)+\(\frac{1}{2}\)-...-\(\frac{1}{49}\)+\(\frac{1}{49}\)-\(\frac{1}{50}\)
=>1-\(\frac{1}{50}\)<1 nên\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+...+\(\frac{1}{50^2}\)<1
vậy ...........................
BD
0
Ta có : 1/2^2<1/1.2
1/3^2 < 1/2.3
1/4^2<1/3.4
................
.............
1/50^2<1/49.50
=> 1/2^2+1/3^2+1/4^2+1/5^2+.....+1/50^2 < 1/1.2+1/2.3+1/3.4+....+1/49.50
=> 1/2^2+1/3^2+1/4^2+1/5^2+.....+1/50^2 < 1-1/50
=> 1/2^2+1/3^2+1/4^2+1/5^2+.....+1/50^2 < 49/50 < 1
Vậy 1/2^2+1/3^2+1/4^2+1/5^2+.....+1/50^2 < 1